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Examples of iterated maps starting from simple initial conditions. At each step there is a number x between 0 and 1 that is updated by applying a fixed mapping. The four mappings considered here are given above both as formulas and in terms of plots. The pictures at the top of the page show the base 2 digit sequences of successive numbers obtained by iterating this mapping, while the pictures in the middle of the page plot the sizes of these numbers. In all cases, the initial conditions consist of the number 1/2—which has a very simple digit sequence. Yet despite this simplicity, cases (a) and (b) show considerable complexity in both the digit sequences and the sizes of the numbers produced (compare page 122). In case (c), the digit sequences are complicated but the sizes of the numbers tend rapidly to zero. And finally, in case (d), neither the digit sequences nor the sizes of numbers are anything but trivial. Note that in the pictures above each horizontal row of digits corresponds to a number, and that digits further to the left contribute progressively more to the size of this number.

From Stephen Wolfram: A New Kind of Science [citation]